M04CG1.1.3a Recognize a line of symmetry in a two
... 3. Least Complex Level: • Content target: Identify when a flat surface (2-D) is separated into equal parts. • Example: Use a simple shape that is easy to identify for sameness when divided. Use only one type of shape per set of examples and no other changes in attributes across shape (color, size, ...
... 3. Least Complex Level: • Content target: Identify when a flat surface (2-D) is separated into equal parts. • Example: Use a simple shape that is easy to identify for sameness when divided. Use only one type of shape per set of examples and no other changes in attributes across shape (color, size, ...
Tutorial 12 - School of Mathematics and Statistics, University of Sydney
... solutions. Replacing f by −f if need be, we may suppose that f (c) > f (a). The restriction of f to the compact interval [a, b] must achieve a maximum value M ≥ f (c) on [a, b]. Suppose that f (d) = M , where d ∈ (a, b). By the assumption about f there must be two solutions of f (x) = M + 1. Choose ...
... solutions. Replacing f by −f if need be, we may suppose that f (c) > f (a). The restriction of f to the compact interval [a, b] must achieve a maximum value M ≥ f (c) on [a, b]. Suppose that f (d) = M , where d ∈ (a, b). By the assumption about f there must be two solutions of f (x) = M + 1. Choose ...
Lecture 3 - Stony Brook Mathematics
... point. Since X is Hausdor↵, we can find two disjoint open sets U and V with x 2 U and y 2 V . Clearly, V ✓ X \ {x}; this shows that X \ {x} is a union of open sets, and therefore open. Example 3.2. Let Y be a subset of a topological space X. Then a set A ✓ Y is closed in the subspace topology on Y i ...
... point. Since X is Hausdor↵, we can find two disjoint open sets U and V with x 2 U and y 2 V . Clearly, V ✓ X \ {x}; this shows that X \ {x} is a union of open sets, and therefore open. Example 3.2. Let Y be a subset of a topological space X. Then a set A ✓ Y is closed in the subspace topology on Y i ...
Math 54 - Lecture 16: Compact Hausdorff Spaces, Products of
... the latter set, which we call U , is open. We need to show that this nbhd But for any z ∈ K, there is a Vyi such that z ∈ Vyi . But then z 6∈ Uyi , so that z 6∈ U . Thus U contains no elements of K, and U ⊂ X − K. The following theorem is remarkably useful when dealing with compact Hausdorff spaces. ...
... the latter set, which we call U , is open. We need to show that this nbhd But for any z ∈ K, there is a Vyi such that z ∈ Vyi . But then z 6∈ Uyi , so that z 6∈ U . Thus U contains no elements of K, and U ⊂ X − K. The following theorem is remarkably useful when dealing with compact Hausdorff spaces. ...
Section 29. Local Compactness - Faculty
... which in turn is contained in compact subspace [x1 − 1, x1 + 1] × [x2 − 1, x2 + 1] × · · · × [xn − 1, xn + 1]. However, Rω = R × R × · · · under the product topology is not locally compact. Recall that basis elements for the product topology are of the form B = (a1 , b1 ) × (a1, a2 ) × · · · × (an , ...
... which in turn is contained in compact subspace [x1 − 1, x1 + 1] × [x2 − 1, x2 + 1] × · · · × [xn − 1, xn + 1]. However, Rω = R × R × · · · under the product topology is not locally compact. Recall that basis elements for the product topology are of the form B = (a1 , b1 ) × (a1, a2 ) × · · · × (an , ...
§2.1. Topological Spaces Let X be a set. A family T of subsets of X is
... converges to x ∈ X iff each point in X other than x appears in the sequence at most finitely many times. (c) Let Z have the cofinite topology. Then the sequence {1, 2, 3, . . . } converges to each point of Z. 1.5. Theorem. If S is a subset of a topological space X and ...
... converges to x ∈ X iff each point in X other than x appears in the sequence at most finitely many times. (c) Let Z have the cofinite topology. Then the sequence {1, 2, 3, . . . } converges to each point of Z. 1.5. Theorem. If S is a subset of a topological space X and ...
RESULT ON VARIATIONAL INEQUALITY PROBLEM 1
... discussed by Gwinner [1], which is, an infinite dimensional version of the Walras excess demand theorem (see also Zeidler [8]), as follows: Theorem 1.1. Let A and B be nonempty compact convex subsets of Hausdorff locally convex topological vector spaces X and Y, respectively. Let f : A × B → R be co ...
... discussed by Gwinner [1], which is, an infinite dimensional version of the Walras excess demand theorem (see also Zeidler [8]), as follows: Theorem 1.1. Let A and B be nonempty compact convex subsets of Hausdorff locally convex topological vector spaces X and Y, respectively. Let f : A × B → R be co ...
Introduction and Table of Contents
... We all learnt Euclidean Geometry at school. This is the geometry that Euclid and his co-workers developed over two thousand years ago and it has to do with points, lines, circles and simple shapes like triangles and parallelograms. Euclid’s methods were axiomatic. He wrote down a number of “intuitiv ...
... We all learnt Euclidean Geometry at school. This is the geometry that Euclid and his co-workers developed over two thousand years ago and it has to do with points, lines, circles and simple shapes like triangles and parallelograms. Euclid’s methods were axiomatic. He wrote down a number of “intuitiv ...
p. 1 Math 490 Notes 12 More About Product Spaces and
... iv) j : (0, ∞) → (a, ∞) defined by j(x) = x + a; v) k : (a, ∞) → (−∞, −a) defined by k(x) = −x. These examples reveal that bounded open intervals, open rays, and R itself are all homeomorphic to one another (with topologies as in the examples). The function in (ii) can be used to establish a homeomo ...
... iv) j : (0, ∞) → (a, ∞) defined by j(x) = x + a; v) k : (a, ∞) → (−∞, −a) defined by k(x) = −x. These examples reveal that bounded open intervals, open rays, and R itself are all homeomorphic to one another (with topologies as in the examples). The function in (ii) can be used to establish a homeomo ...
Part II - Cornell Math
... condition (ii) since the union of finite collection of compact sets is compact. Since X 0 − W is compact in a Hausdorff space X, it is closed, and so W = X \ (X 0 − W) is open in X. Thus, we have V ∩ W ⊂ T . (b) Given U ⊂ T . If U ⊂ X, then X ∩ U = U is open in X by condition (i). If U 1 X, then X 0 ...
... condition (ii) since the union of finite collection of compact sets is compact. Since X 0 − W is compact in a Hausdorff space X, it is closed, and so W = X \ (X 0 − W) is open in X. Thus, we have V ∩ W ⊂ T . (b) Given U ⊂ T . If U ⊂ X, then X ∩ U = U is open in X by condition (i). If U 1 X, then X 0 ...
